### Isosceles Triangles

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

### Route to Infinity

Can you describe this route to infinity? Where will the arrows take you next?

### Eight Hidden Squares

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

# Lost

##### Stage: 3 Challenge Level:

We received a number of solutions to this problem. Stephen from Manly Selective Campus got very close to the most efficient strategy:

To explain the solution, I will use an example of this problem.
Say the giraffe is at (6,8).
The first step would be to enter the co-ordinates (0,0).
It will then say that you are 14 blocks away from the giraffe, because the two co-ordinates will add up to the number of blocks you are away from the giraffe.
The next step would be to then type in the co-ordinates (7,7).
It will then say that you are 2 blocks away from the giraffe.
So, that leaves you with 2 possibilities: (6,8) and (8,6).
It is a simple matter then to try the two and hope that the one that you say first is right.

To be sure you will not need more than 3 guesses you will need to adopt the strategy worked out by a number of pupils from St Hilda's Anglican School for Girls.

Gloria and Sneha suggested:

1) Start off at any corner eg. (0,0)
A diagonal of possibilities will form.
2) Click on one of the two edges of the diagonal as this gives only one possible solution.
If you dont click on the edge, you could have two possibilities.
3) Locate the point on the diagonal which is 'n' blocks away from the edge of the diagonal.
Snap!!

Emily wrote:

Your first point must be on one of the 4 corners (eg. (9,9) or (0,9) etc).
You should then plot out where the giraffe could possibly be using the information you have
(eg. if you started from (9,9) and the giraffe is 2 blocks away, it could be at either (9,7), (8,8) or (7,9))
Choose one of the extremes or outer co-ordinates (either (9,7) or (7,9)) and, from the information you are given from that search, plot where the giraffe could be.
One of these points will overlap with one of the points from the other search and this is where the giraffe is (eg. if you chose (9,7) and the giraffe was 4 blocks away, it would be at (7,9)).

Ailie and Sophie summarised the strategy very clearly:

Choose one of the 4 corners (9,9) (0,0) (9,0) (0,9).
Then, out of the possibilities choose one that is on an edge.
After you do that there will be only one possibility where the giraffe is which will be one of the possible co-ordinates from before.

We also received correct solutions from Tessa, Katharine and Alarna, also pupils from St Hilda's Anglican School for Girls. Well done to you all.